In-class Exercise 5: Modeling the Spatial Variation of the Explanatory Factors of Water Point Status using Geographically Weighted Logistic Regression

Author

Zheng Yimin

Published

December 17, 2022

Setting the scene

  • To build an exploratory model to discover factors affecting water point in Osun State, Nigeria

  • Study area: Osun State, Nigeria

  • Data sets used:

    • Osun.rds, which contains LGA boundaries of Osun state. It is in sf polygon data frame, and

    • Osun_wp_sf.rds, which contains water points within Osun state. It is in sf point data frame

Model Variables

  • Dependent variable: Water point status (i.e. functional/non-functional)

  • Independent variables:

    • distance_to_primary_road (continuous),

    • distance_to_secondary_road (continuous),

    • distance_to_tertiary_road (continuous),

    • distance_to_city (continuous),

    • distance_to_town (continuous),

    • water_point_population (continuous),

    • local_population_11km (continuous),

    • usage_capacity (categorical),

    • is_urban (categorical),

    • water_source_clean (categorical)

IntroductionLogistic Regression Model

We will take a look at the theory of Logistic regression model before performing our analysis.

Simple Logistic Regression Model

Let’s take a look at the formula for a simple logistic regression model:

Multiple Logistic Regression model

Assumptions of logistic regression

  • Logistic regression does not assume a linear relationship between dependent and independent variables.

  • For binary logistic regression, the dependent variable must be a dichotomy (2 categories).

  • The independent variables need not be interval, nor normally distributed, nor linearly related, nor of equal variance within each group.

  • The categories (groups) must be mutually exclusive and exhaustive; a case can only be in one group and every case must be a member of one of the groups.

Data requirements of logistic regression

  • Large samples are required

  • It does not use least squares method, but instead, it uses an algorithm called maximum likelihood (expectation of the probability)

Getting started

The code chunks below installs and launches the R packages into R environment. The below packages mentioned are specifically used for this exercise to support our analysis.

blorr - for error matrix computations

skimr, funModeling - for exploratory data analysis

caret - commonly for machine learning purposes, and it is used in this exercise to compute the error matrix for comparison with the output from blorr.

pacman::p_load(sf, tidyverse, funModeling, blorr, corrplot, spdep, GWmodel, tmap, skimr, caret)

Data import

We will now import the required dataset into the R environment. Two datasets will be used.

Osun <- read_rds("rds/Osun.rds")
Osun_wp_sf <- read_rds("rds/Osun_wp_sf.rds")

We will also use funModeling to take a look at the data records under the status column.

Osun_wp_sf %>% freq(input="status")
Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
  Please report the issue at <]8;;https://github.com/pablo14/funModeling/issueshttps://github.com/pablo14/funModeling/issues]8;;>.

  status frequency percentage cumulative_perc
1   TRUE      2642       55.5            55.5
2  FALSE      2118       44.5           100.0
tmap_mode("view")
tmap mode set to interactive viewing
tm_shape(Osun) +
  # tmap_options(check.and.fix=TRUE) +
  tm_polygons(alpha = 0.4) +
  tm_shape(Osun_wp_sf) +
  tm_dots(col="status", alpha = 0.6) +
  tm_view(set.zoom.limits = c(9,12))

It is interesting to note here that the FALSE variables are closely clustered in some of the regions.

Exploratory Data Analysis (EDA)

Note: Regression models are very sensitive to missing values. Hence, we should remove all missing values from the data set before continuing with our analysis.

We will now use skimr to do EDA. It provides a tabular format of the summary statistics of the variables that we will be looking into.

Summary statistics with skimr

Osun_wp_sf %>% skim()
Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.
Data summary
Name Piped data
Number of rows 4760
Number of columns 75
_______________________
Column type frequency:
character 47
logical 5
numeric 23
________________________
Group variables None

Variable type: character

skim_variable n_missing complete_rate min max empty n_unique whitespace
source 0 1.00 5 44 0 2 0
report_date 0 1.00 22 22 0 42 0
status_id 0 1.00 2 7 0 3 0
water_source_clean 0 1.00 8 22 0 3 0
water_source_category 0 1.00 4 6 0 2 0
water_tech_clean 24 0.99 9 23 0 3 0
water_tech_category 24 0.99 9 15 0 2 0
facility_type 0 1.00 8 8 0 1 0
clean_country_name 0 1.00 7 7 0 1 0
clean_adm1 0 1.00 3 5 0 5 0
clean_adm2 0 1.00 3 14 0 35 0
clean_adm3 4760 0.00 NA NA 0 0 0
clean_adm4 4760 0.00 NA NA 0 0 0
installer 4760 0.00 NA NA 0 0 0
management_clean 1573 0.67 5 37 0 7 0
status_clean 0 1.00 9 32 0 7 0
pay 0 1.00 2 39 0 7 0
fecal_coliform_presence 4760 0.00 NA NA 0 0 0
subjective_quality 0 1.00 18 20 0 4 0
activity_id 4757 0.00 36 36 0 3 0
scheme_id 4760 0.00 NA NA 0 0 0
wpdx_id 0 1.00 12 12 0 4760 0
notes 0 1.00 2 96 0 3502 0
orig_lnk 4757 0.00 84 84 0 1 0
photo_lnk 41 0.99 84 84 0 4719 0
country_id 0 1.00 2 2 0 1 0
data_lnk 0 1.00 79 96 0 2 0
water_point_history 0 1.00 142 834 0 4750 0
clean_country_id 0 1.00 3 3 0 1 0
country_name 0 1.00 7 7 0 1 0
water_source 0 1.00 8 30 0 4 0
water_tech 0 1.00 5 37 0 20 0
adm2 0 1.00 3 14 0 33 0
adm3 4760 0.00 NA NA 0 0 0
management 1573 0.67 5 47 0 7 0
adm1 0 1.00 4 5 0 4 0
New Georeferenced Column 0 1.00 16 35 0 4760 0
lat_lon_deg 0 1.00 13 32 0 4760 0
public_data_source 0 1.00 84 102 0 2 0
converted 0 1.00 53 53 0 1 0
created_timestamp 0 1.00 22 22 0 2 0
updated_timestamp 0 1.00 22 22 0 2 0
Geometry 0 1.00 33 37 0 4760 0
ADM2_EN 0 1.00 3 14 0 30 0
ADM2_PCODE 0 1.00 8 8 0 30 0
ADM1_EN 0 1.00 4 4 0 1 0
ADM1_PCODE 0 1.00 5 5 0 1 0

Variable type: logical

skim_variable n_missing complete_rate mean count
rehab_year 4760 0 NaN :
rehabilitator 4760 0 NaN :
is_urban 0 1 0.39 FAL: 2884, TRU: 1876
latest_record 0 1 1.00 TRU: 4760
status 0 1 0.56 TRU: 2642, FAL: 2118

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
row_id 0 1.00 68550.48 10216.94 49601.00 66874.75 68244.50 69562.25 471319.00 ▇▁▁▁▁
lat_deg 0 1.00 7.68 0.22 7.06 7.51 7.71 7.88 8.06 ▁▂▇▇▇
lon_deg 0 1.00 4.54 0.21 4.08 4.36 4.56 4.71 5.06 ▃▆▇▇▂
install_year 1144 0.76 2008.63 6.04 1917.00 2006.00 2010.00 2013.00 2015.00 ▁▁▁▁▇
fecal_coliform_value 4760 0.00 NaN NA NA NA NA NA NA
distance_to_primary_road 0 1.00 5021.53 5648.34 0.01 719.36 2972.78 7314.73 26909.86 ▇▂▁▁▁
distance_to_secondary_road 0 1.00 3750.47 3938.63 0.15 460.90 2554.25 5791.94 19559.48 ▇▃▁▁▁
distance_to_tertiary_road 0 1.00 1259.28 1680.04 0.02 121.25 521.77 1834.42 10966.27 ▇▂▁▁▁
distance_to_city 0 1.00 16663.99 10960.82 53.05 7930.75 15030.41 24255.75 47934.34 ▇▇▆▃▁
distance_to_town 0 1.00 16726.59 12452.65 30.00 6876.92 12204.53 27739.46 44020.64 ▇▅▃▃▂
rehab_priority 2654 0.44 489.33 1658.81 0.00 7.00 91.50 376.25 29697.00 ▇▁▁▁▁
water_point_population 4 1.00 513.58 1458.92 0.00 14.00 119.00 433.25 29697.00 ▇▁▁▁▁
local_population_1km 4 1.00 2727.16 4189.46 0.00 176.00 1032.00 3717.00 36118.00 ▇▁▁▁▁
crucialness_score 798 0.83 0.26 0.28 0.00 0.07 0.15 0.35 1.00 ▇▃▁▁▁
pressure_score 798 0.83 1.46 4.16 0.00 0.12 0.41 1.24 93.69 ▇▁▁▁▁
usage_capacity 0 1.00 560.74 338.46 300.00 300.00 300.00 1000.00 1000.00 ▇▁▁▁▅
days_since_report 0 1.00 2692.69 41.92 1483.00 2688.00 2693.00 2700.00 4645.00 ▁▇▁▁▁
staleness_score 0 1.00 42.80 0.58 23.13 42.70 42.79 42.86 62.66 ▁▁▇▁▁
location_id 0 1.00 235865.49 6657.60 23741.00 230638.75 236199.50 240061.25 267454.00 ▁▁▁▁▇
cluster_size 0 1.00 1.05 0.25 1.00 1.00 1.00 1.00 4.00 ▇▁▁▁▁
lat_deg_original 4760 0.00 NaN NA NA NA NA NA NA
lon_deg_original 4760 0.00 NaN NA NA NA NA NA NA
count 0 1.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 ▁▁▇▁▁

We can see from the second table of the summary statistics that some of the columns contains many missing values (e.g. Clean_ADM3). Hence, we should remove columns that have many missing values. Also, as seen from the third table, there are also some variables with excessive missing values (e.g. install_year) and these should be removed as well.

We can also notice in the third table, the water_point_population and local_population_11km, have 4 missing values respectively, and they are all referring to the same data records. Hence, we will run the code chunk below to extract the variables used in this exercise, as well as remove the 4 data records that are missing in both the water_point_population and local_population_11km columns. usage_capacity has also been changed to factor class (with only two levels) from numeric class.

Osun_wp_sf_clean <- Osun_wp_sf %>%
  filter_at(vars(status, distance_to_primary_road, 
                 distance_to_secondary_road,
                 distance_to_tertiary_road,
                 distance_to_city, 
                 distance_to_town,
                 water_point_population,
                 local_population_1km,
                 usage_capacity,
                 is_urban,
                 water_source_clean),
            all_vars(!is.na(.))) %>%
  mutate(usage_capacity = as.factor(usage_capacity))
Osun_wp <- Osun_wp_sf_clean %>%
  select(c(7,35:39,42:43, 46:47,57)) %>%
  st_set_geometry(NULL)

Correlation Analysis

We will run the code chunk below to perform correlation analysis.

cluster_vars.cor = cor(Osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
               lower = "ellipse",
               upper = "number",
               tl.pos = "lt",
               diag = "l",
               tl.col = "black")

We can see there is no sign of multi-collinearity in the correlation plot above (i.e. no 2 independent variables have correlation of higher than 0.85).

Building a logistic Regression Model

We will now construct our logistic regression model.

model <- glm(status ~ distance_to_primary_road + 
                 distance_to_secondary_road +
                 distance_to_tertiary_road +
                 distance_to_city +
                 distance_to_town + 
                 water_point_population+ 
                 local_population_1km +
                 usage_capacity +
                 is_urban +
                 water_source_clean, 
             data = Osun_wp_sf_clean, 
             family = binomial(link="logit"))

We will run blr_regress() to generate the output, based on our code chunk above.

blr_regress(model)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4744           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3887        0.1124      3.4588       5e-04 
        distance_to_primary_road            1      0.0000        0.0000     -0.7153      0.4744 
       distance_to_secondary_road           1      0.0000        0.0000     -0.5530      0.5802 
       distance_to_tertiary_road            1      1e-04         0.0000      4.6708      0.0000 
            distance_to_city                1      0.0000        0.0000     -4.7574      0.0000 
            distance_to_town                1      0.0000        0.0000     -4.9170      0.0000 
         water_point_population             1      -5e-04        0.0000    -11.3686      0.0000 
          local_population_1km              1      3e-04         0.0000     19.2953      0.0000 
           usage_capacity1000               1     -0.6230        0.0697     -8.9366      0.0000 
              is_urbanTRUE                  1     -0.2971        0.0819     -3.6294       3e-04 
water_source_cleanProtected Shallow Well    1      0.5040        0.0857      5.8783      0.0000 
   water_source_cleanProtected Spring       1      1.2882        0.4388      2.9359      0.0033 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7347          Somers' D        0.4693   
% Discordant          0.2653          Gamma            0.4693   
% Tied                0.0000          Tau-a            0.2318   
Pairs                5585188          c                0.7347   
---------------------------------------------------------------

Interpretation of logistic regression

Interpretation should only be done after analysing the respective p-values of each of the variables.

For categorical variables, a positive p-value implies an above average correlation and a negative p-value implies a below average correlation.

For continuous variables, a positive p-value implies a direct correlation and a negative value implies an inverse correlation, while the magnitude of the p-value gives the strength of the correlation.

We can notice in the above summary, there are two variables (distance_to_primary_road & distance_to_secondary_road) with p value greater than the confidence interval of 0.05. Hence, we will exclude these 2 variables as they are not statistically significant.

In the code chunk below, blr_confusion_matrix() of blorr package is used to compute the confusion matrix of the estimated outcomes by using 0.5 as the cutoff value.

blr_confusion_matrix(model, cutoff = 0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1301  738
         1   813 1904

                Accuracy : 0.6739 
     No Information Rate : 0.4445 

                   Kappa : 0.3373 

McNemars's Test P-Value  : 0.0602 

             Sensitivity : 0.7207 
             Specificity : 0.6154 
          Pos Pred Value : 0.7008 
          Neg Pred Value : 0.6381 
              Prevalence : 0.5555 
          Detection Rate : 0.4003 
    Detection Prevalence : 0.5713 
       Balanced Accuracy : 0.6680 
               Precision : 0.7008 
                  Recall : 0.7207 

        'Positive' Class : 1

The validity of a cut-off is measured using sensitivity, specificity and accuracy.

  • Sensitivity: The % of correctly classified events out of all events = TP (TP + FN)

    • The value of true positives is higher than the value of true negatives.
  • Specificity: The % of correctly classified non-events out of all non-events = TN / (TN+ FP)

  • Accuracy: The % of correctly classified observations over all observations = (TP + TN) / (TP + TN + FP + FN).

  • False positive rate = FP / (TN + FP)

    • We can see that the accuracy here is 0.6739, which is a relatively high value.

Building Geographically Weighted Logistic Regression (gwLR) models

Converting from sf to sp data frame

We will first convert Osun_wp_sf_clean to Spatial Point data frame (Osun_wp_sp). The clean version is used

Osun_wp_sp <- Osun_wp_sf_clean %>%
  select(c(status, 
                 distance_to_primary_road, 
                 distance_to_secondary_road,
                 distance_to_tertiary_road,
                 distance_to_city, 
                 distance_to_town,
                 water_point_population,
                 local_population_1km,
                 usage_capacity,
                 is_urban,
                 water_source_clean)) %>%
  as_Spatial()
Osun_wp_sp
class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 11
names       : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, usage_capacity, is_urban, water_source_clean 
min values  :      0,        0.014461356813335,          0.152195902540837,         0.017815121653488, 53.0461399623541, 30.0019777713073,                      0,                    0,           1000,        0,           Borehole 
max values  :      1,         26909.8616132094,           19559.4793799085,          10966.2705628969,  47934.343603562, 44020.6393368124,                  29697,                36118,            300,        1,   Protected Spring 

Building Fixed Bandwidth GWR Model

Computing fixed bandwidth

bw.fixed <- bw.ggwr(status ~ distance_to_primary_road + 
                 distance_to_secondary_road +
                 distance_to_tertiary_road +
                 distance_to_city +
                 distance_to_town + 
                 water_point_population+ 
                 local_population_1km +
                 usage_capacity +
                 is_urban +
                 water_source_clean, 
             data = Osun_wp_sp,
             family="binomial",
             approach="AIC",
             kernel="gaussian",
             adaptive=FALSE,
             longlat=FALSE)
Take a cup of tea and have a break, it will take a few minutes.
          -----A kind suggestion from GWmodel development group
 Iteration    Log-Likelihood:(With bandwidth:  95768.67 )
=========================
       0        -2889 
       1        -2836 
       2        -2830 
       3        -2829 
       4        -2829 
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Fixed bandwidth: 95768.67 AICc value: 5684.357 
 Iteration    Log-Likelihood:(With bandwidth:  59200.13 )
=========================
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Fixed bandwidth: 59200.13 AICc value: 5646.785 
 Iteration    Log-Likelihood:(With bandwidth:  36599.53 )
=========================
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Fixed bandwidth: 36599.53 AICc value: 5575.148 
 Iteration    Log-Likelihood:(With bandwidth:  22631.59 )
=========================
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Fixed bandwidth: 22631.59 AICc value: 5466.883 
 Iteration    Log-Likelihood:(With bandwidth:  13998.93 )
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Fixed bandwidth: 13998.93 AICc value: 5324.578 
 Iteration    Log-Likelihood:(With bandwidth:  8663.649 )
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Fixed bandwidth: 8663.649 AICc value: 5163.61 
 Iteration    Log-Likelihood:(With bandwidth:  5366.266 )
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Fixed bandwidth: 5366.266 AICc value: 4990.587 
 Iteration    Log-Likelihood:(With bandwidth:  3328.371 )
=========================
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Fixed bandwidth: 3328.371 AICc value: 4798.288 
 Iteration    Log-Likelihood:(With bandwidth:  2068.882 )
=========================
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Fixed bandwidth: 2068.882 AICc value: 4837.017 
 Iteration    Log-Likelihood:(With bandwidth:  4106.777 )
=========================
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Fixed bandwidth: 4106.777 AICc value: 4873.161 
 Iteration    Log-Likelihood:(With bandwidth:  2847.289 )
=========================
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Fixed bandwidth: 2847.289 AICc value: 4768.192 
 Iteration    Log-Likelihood:(With bandwidth:  2549.964 )
=========================
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Fixed bandwidth: 2549.964 AICc value: 4762.212 
 Iteration    Log-Likelihood:(With bandwidth:  2366.207 )
=========================
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Fixed bandwidth: 2366.207 AICc value: 4773.081 
 Iteration    Log-Likelihood:(With bandwidth:  2663.532 )
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Fixed bandwidth: 2663.532 AICc value: 4762.568 
 Iteration    Log-Likelihood:(With bandwidth:  2479.775 )
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Fixed bandwidth: 2479.775 AICc value: 4764.294 
 Iteration    Log-Likelihood:(With bandwidth:  2593.343 )
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Fixed bandwidth: 2593.343 AICc value: 4761.813 
 Iteration    Log-Likelihood:(With bandwidth:  2620.153 )
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Fixed bandwidth: 2620.153 AICc value: 4761.89 
 Iteration    Log-Likelihood:(With bandwidth:  2576.774 )
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Fixed bandwidth: 2576.774 AICc value: 4761.889 
 Iteration    Log-Likelihood:(With bandwidth:  2603.584 )
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Fixed bandwidth: 2603.584 AICc value: 4761.813 
 Iteration    Log-Likelihood:(With bandwidth:  2609.913 )
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Fixed bandwidth: 2609.913 AICc value: 4761.831 
 Iteration    Log-Likelihood:(With bandwidth:  2599.672 )
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Fixed bandwidth: 2599.672 AICc value: 4761.809 
 Iteration    Log-Likelihood:(With bandwidth:  2597.255 )
=========================
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Fixed bandwidth: 2597.255 AICc value: 4761.809 
bw.fixed
[1] 2599.672

We can see that the recommended maximum bandwidth is 2,599.672m. We will input this value into the code chunk below to compute the geographically weighted logistic regression.

gwlr.fixed <- ggwr.basic(status ~ 
                 distance_to_primary_road + 
                 distance_to_secondary_road +
                 distance_to_tertiary_road +
                 distance_to_city +
                 distance_to_town + 
                 water_point_population+ 
                 local_population_1km +
                 usage_capacity +
                 is_urban +
                 water_source_clean,
              data = Osun_wp_sp,
              bw=bw.fixed,
              family="binomial",
              kernel = "gaussian",
              adaptive=FALSE,
              longlat=FALSE)
 Iteration    Log-Likelihood
=========================
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       1        -1676 
       2        -1526 
       3        -1443 
       4        -1405 
       5        -1405 
gwlr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-17 21:26:23 
   Call:
   ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road + 
    distance_to_tertiary_road + distance_to_city + distance_to_town + 
    water_point_population + local_population_1km + usage_capacity + 
    is_urban + water_source_clean, data = Osun_wp_sp, bw = bw.fixed, 
    family = "binomial", kernel = "gaussian", adaptive = FALSE, 
    longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-124.555    -1.755     1.072     1.742    34.333  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.887e-01  1.124e-01   3.459 0.000543
distance_to_primary_road                 -4.642e-06  6.490e-06  -0.715 0.474422
distance_to_secondary_road               -5.143e-06  9.299e-06  -0.553 0.580230
distance_to_tertiary_road                 9.683e-05  2.073e-05   4.671 3.00e-06
distance_to_city                         -1.686e-05  3.544e-06  -4.757 1.96e-06
distance_to_town                         -1.480e-05  3.009e-06  -4.917 8.79e-07
water_point_population                   -5.097e-04  4.484e-05 -11.369  < 2e-16
local_population_1km                      3.451e-04  1.788e-05  19.295  < 2e-16
usage_capacity1000                       -6.230e-01  6.972e-02  -8.937  < 2e-16
is_urbanTRUE                             -2.971e-01  8.185e-02  -3.629 0.000284
water_source_cleanProtected Shallow Well  5.040e-01  8.574e-02   5.878 4.14e-09
water_source_cleanProtected Spring        1.288e+00  4.388e-01   2.936 0.003325
                                            
Intercept                                ***
distance_to_primary_road                    
distance_to_secondary_road                  
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
usage_capacity1000                       ***
is_urbanTRUE                             ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.0  on 4744  degrees of freedom
AIC: 5712

Number of Fisher Scoring iterations: 5


 AICc:  5712.099
 Pseudo R-square value:  0.1295351
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2599.672 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -8.7228e+02 -4.9955e+00  1.7600e+00
   distance_to_primary_road                 -1.9389e-02 -4.8031e-04  2.9618e-05
   distance_to_secondary_road               -1.5921e-02 -3.7551e-04  1.2317e-04
   distance_to_tertiary_road                -1.5618e-02 -4.2368e-04  7.6179e-05
   distance_to_city                         -1.8416e-02 -5.6217e-04 -1.2726e-04
   distance_to_town                         -2.2411e-02 -5.7283e-04 -1.5155e-04
   water_point_population                   -5.2208e-02 -2.2767e-03 -9.8875e-04
   local_population_1km                     -1.2698e-01  4.9952e-04  1.0638e-03
   usage_capacity1000                       -2.0772e+01 -9.7231e-01 -4.1592e-01
   is_urbanTRUE                             -1.9790e+02 -4.2908e+00 -1.6864e+00
   water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01  5.3340e-01
   water_source_cleanProtected.Spring       -5.2235e+02 -5.5977e+00  2.5441e+00
                                                3rd Qu.      Max.
   Intercept                                 1.2763e+01 1073.2154
   distance_to_primary_road                  4.8443e-04    0.0142
   distance_to_secondary_road                6.0692e-04    0.0258
   distance_to_tertiary_road                 6.6814e-04    0.0128
   distance_to_city                          2.3718e-04    0.0150
   distance_to_town                          1.9271e-04    0.0224
   water_point_population                    5.0102e-04    0.1309
   local_population_1km                      1.8157e-03    0.0392
   usage_capacity1000                        3.0322e-01    5.9281
   is_urbanTRUE                              1.2841e+00  744.3097
   water_source_cleanProtected.Shallow.Well  1.7849e+00   67.6343
   water_source_cleanProtected.Spring        6.7663e+00  317.4123
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2795.084 
   AIC : 4414.606 
   AICc : 4747.423 
   Pseudo R-square value:  0.5722559 

   ***********************************************************************
   Program stops at: 2022-12-17 21:27:36 

Here, the AIC value is computed and compared with non-geographic weighted logistic regression AIC value. We can see that the AIC value for geographic weighted logistic regression is lower (4,414.606) in this case. This shows that the model is actually improved since the AIC value has decreased.

Now we will use caret package to analyse the aspects which has resulted in the model’s improvement.

Model Assessment

Converting SDF into sf data frame

To assess the performance of the gwLR, firstly, we will convert the SDF object in as data frame by using the code chunk below.

gwr.fixed <- as.data.frame(gwlr.fixed$SDF)

Next, we will label yhat values greater than or equal to 0.5 into 1 and else, 0. The result of the logic comparison operation will be saved into a field called most. most is a logical function that has only True or False values, which matches with the observed values.

gwr.fixed <- gwr.fixed %>%
  mutate(most = ifelse(
    gwr.fixed$yhat >= 0.5, T, F))

The code chunk below generates another confusion matrix, which will be used in comparison to the confusion matrix generated at an earlier step (global version).

gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data=gwr.fixed$most, 
                      positive="TRUE",
                      reference = gwr.fixed$y)
CM
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1824  263
     TRUE    290 2379
                                          
               Accuracy : 0.8837          
                 95% CI : (0.8743, 0.8927)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7642          
                                          
 Mcnemar's Test P-Value : 0.2689          
                                          
            Sensitivity : 0.9005          
            Specificity : 0.8628          
         Pos Pred Value : 0.8913          
         Neg Pred Value : 0.8740          
             Prevalence : 0.5555          
         Detection Rate : 0.5002          
   Detection Prevalence : 0.5612          
      Balanced Accuracy : 0.8816          
                                          
       'Positive' Class : TRUE            
                                          

As we can see, the structure of both confusion matrix are more or less the same. However, the accuracy has increased to 0.8837, which is a significant increase from the global version of 0.6739. There is no change in the variables, except that one model is non-geographic weighted, and the other is geographic weighted.

The true negatives has also increased from 0.6154 to 0.9005. This means that we can explain the non-functional water points better now. In order to better manage the water points, we should look more into the local neighbourhoods more, rather than at a global scale (i.e. consider local factors instead).

We can thus see that the geographic weighted model is a better model.

Visualising gwLR

Osun_wp_sf_selected <- Osun_wp_sf_clean %>%
  select(c(ADM2_EN, ADM2_PCODE, ADM1_EN, ADM1_PCODE, status))
gwr_sf.fixed <- cbind(Osun_wp_sf_selected, gwr.fixed)

The code chunk below is used to create an interactive point symbol map.

tmap_mode("view")
tmap mode set to interactive viewing
prob_T <- tm_shape(Osun) +
  tm_polygons(alpha=0.1) +
  tm_text(text="ADM2_EN") +
  tm_shape(gwr_sf.fixed) +
  tm_dots(col="yhat",
          border.col="gray60",
          border.lwd=1) +
  tm_view(set.zoom.limits = c(8,14))
prob_T

From the map above, we can note that some of the states had more clusters of water points than as compared to the others. This is more prevalent in states like Ifelodun, and etc.

# tertiary_TV <- tm_shape(Osun) +
  #tm_polygons(alpha=0.1) +
  #tm_shape(gwr_sf.fixed) +
  #tm_dots(col="distance_to_tertiary_road_TV",border.col="gray60",border.lwd=1) +
  #tm_view(set.zoom.limits=c(8,14))
# tmap_arrange(tertiary_S,tertiary_TV,asp=1,ncol=2,sync=TRUE)

Building Revised Geographically Weighted Logistic Regression (gwLR) models

As we have seen earlier on, there are 2 variables that are not statistically significant. We will proceed to drop these 2 variables and compute the fixed bandwidth geographically weighted logistic regression model.

Converting from sf to sp data frame

We will first convert Osun_wp_sf_clean to Spatial Point data frame (Osun_wp_sp_new). The clean version is used

Osun_wp_sp_new <- Osun_wp_sf_clean %>%
  select(c(status, 
                 distance_to_tertiary_road,
                 distance_to_city, 
                 distance_to_town,
                 water_point_population,
                 local_population_1km,
                 usage_capacity,
                 is_urban,
                 water_source_clean)) %>%
  as_Spatial()
Osun_wp_sp_new
class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 9
names       : status, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, usage_capacity, is_urban, water_source_clean 
min values  :      0,         0.017815121653488, 53.0461399623541, 30.0019777713073,                      0,                    0,           1000,        0,           Borehole 
max values  :      1,          10966.2705628969,  47934.343603562, 44020.6393368124,                  29697,                36118,            300,        1,   Protected Spring 

Building Fixed Bandwidth GWR Model

Computing fixed bandwidth

bw_new.fixed <- bw.ggwr(status ~ 
                 distance_to_tertiary_road +
                 distance_to_city +
                 distance_to_town + 
                 water_point_population+ 
                 local_population_1km +
                 usage_capacity +
                 is_urban +
                 water_source_clean, 
             data = Osun_wp_sp_new,
             family="binomial",
             approach="AIC",
             kernel="gaussian",
             adaptive=FALSE,
             longlat=FALSE)
Take a cup of tea and have a break, it will take a few minutes.
          -----A kind suggestion from GWmodel development group
 Iteration    Log-Likelihood:(With bandwidth:  95768.67 )
=========================
       0        -2890 
       1        -2837 
       2        -2830 
       3        -2829 
       4        -2829 
       5        -2829 
Fixed bandwidth: 95768.67 AICc value: 5681.18 
 Iteration    Log-Likelihood:(With bandwidth:  59200.13 )
=========================
       0        -2878 
       1        -2820 
       2        -2812 
       3        -2810 
       4        -2810 
       5        -2810 
Fixed bandwidth: 59200.13 AICc value: 5645.901 
 Iteration    Log-Likelihood:(With bandwidth:  36599.53 )
=========================
       0        -2854 
       1        -2790 
       2        -2777 
       3        -2774 
       4        -2774 
       5        -2774 
       6        -2774 
Fixed bandwidth: 36599.53 AICc value: 5585.354 
 Iteration    Log-Likelihood:(With bandwidth:  22631.59 )
=========================
       0        -2810 
       1        -2732 
       2        -2711 
       3        -2707 
       4        -2707 
       5        -2707 
       6        -2707 
Fixed bandwidth: 22631.59 AICc value: 5481.877 
 Iteration    Log-Likelihood:(With bandwidth:  13998.93 )
=========================
       0        -2732 
       1        -2635 
       2        -2604 
       3        -2597 
       4        -2596 
       5        -2596 
       6        -2596 
Fixed bandwidth: 13998.93 AICc value: 5333.718 
 Iteration    Log-Likelihood:(With bandwidth:  8663.649 )
=========================
       0        -2624 
       1        -2502 
       2        -2459 
       3        -2447 
       4        -2446 
       5        -2446 
       6        -2446 
       7        -2446 
Fixed bandwidth: 8663.649 AICc value: 5178.493 
 Iteration    Log-Likelihood:(With bandwidth:  5366.266 )
=========================
       0        -2478 
       1        -2319 
       2        -2250 
       3        -2225 
       4        -2219 
       5        -2219 
       6        -2220 
       7        -2220 
       8        -2220 
       9        -2220 
Fixed bandwidth: 5366.266 AICc value: 5022.016 
 Iteration    Log-Likelihood:(With bandwidth:  3328.371 )
=========================
       0        -2222 
       1        -2002 
       2        -1894 
       3        -1838 
       4        -1818 
       5        -1814 
       6        -1814 
Fixed bandwidth: 3328.371 AICc value: 4827.587 
 Iteration    Log-Likelihood:(With bandwidth:  2068.882 )
=========================
       0        -1837 
       1        -1528 
       2        -1357 
       3        -1261 
       4        -1222 
       5        -1222 
Fixed bandwidth: 2068.882 AICc value: 4772.046 
 Iteration    Log-Likelihood:(With bandwidth:  1290.476 )
=========================
       0        -1403 
       1        -1016 
       2       -807.3 
       3       -680.2 
       4       -680.2 
Fixed bandwidth: 1290.476 AICc value: 5809.734 
 Iteration    Log-Likelihood:(With bandwidth:  2549.964 )
=========================
       0        -2019 
       1        -1753 
       2        -1614 
       3        -1538 
       4        -1506 
       5        -1506 
Fixed bandwidth: 2549.964 AICc value: 4764.056 
 Iteration    Log-Likelihood:(With bandwidth:  2847.289 )
=========================
       0        -2108 
       1        -1862 
       2        -1736 
       3        -1670 
       4        -1644 
       5        -1644 
Fixed bandwidth: 2847.289 AICc value: 4791.834 
 Iteration    Log-Likelihood:(With bandwidth:  2366.207 )
=========================
       0        -1955 
       1        -1675 
       2        -1525 
       3        -1441 
       4        -1407 
       5        -1407 
Fixed bandwidth: 2366.207 AICc value: 4755.524 
 Iteration    Log-Likelihood:(With bandwidth:  2252.639 )
=========================
       0        -1913 
       1        -1623 
       2        -1465 
       3        -1376 
       4        -1341 
       5        -1341 
Fixed bandwidth: 2252.639 AICc value: 4759.188 
 Iteration    Log-Likelihood:(With bandwidth:  2436.396 )
=========================
       0        -1980 
       1        -1706 
       2        -1560 
       3        -1479 
       4        -1446 
       5        -1446 
Fixed bandwidth: 2436.396 AICc value: 4756.675 
 Iteration    Log-Likelihood:(With bandwidth:  2322.828 )
=========================
       0        -1940 
       1        -1656 
       2        -1503 
       3        -1417 
       4        -1382 
       5        -1382 
Fixed bandwidth: 2322.828 AICc value: 4756.471 
 Iteration    Log-Likelihood:(With bandwidth:  2393.017 )
=========================
       0        -1965 
       1        -1687 
       2        -1539 
       3        -1456 
       4        -1422 
       5        -1422 
Fixed bandwidth: 2393.017 AICc value: 4755.57 
 Iteration    Log-Likelihood:(With bandwidth:  2349.638 )
=========================
       0        -1949 
       1        -1668 
       2        -1517 
       3        -1432 
       4        -1398 
       5        -1398 
Fixed bandwidth: 2349.638 AICc value: 4755.753 
 Iteration    Log-Likelihood:(With bandwidth:  2376.448 )
=========================
       0        -1959 
       1        -1680 
       2        -1530 
       3        -1447 
       4        -1413 
       5        -1413 
Fixed bandwidth: 2376.448 AICc value: 4755.48 
 Iteration    Log-Likelihood:(With bandwidth:  2382.777 )
=========================
       0        -1961 
       1        -1683 
       2        -1534 
       3        -1450 
       4        -1416 
       5        -1416 
Fixed bandwidth: 2382.777 AICc value: 4755.491 
 Iteration    Log-Likelihood:(With bandwidth:  2372.536 )
=========================
       0        -1958 
       1        -1678 
       2        -1528 
       3        -1445 
       4        -1411 
       5        -1411 
Fixed bandwidth: 2372.536 AICc value: 4755.488 
 Iteration    Log-Likelihood:(With bandwidth:  2378.865 )
=========================
       0        -1960 
       1        -1681 
       2        -1532 
       3        -1448 
       4        -1414 
       5        -1414 
Fixed bandwidth: 2378.865 AICc value: 4755.481 
 Iteration    Log-Likelihood:(With bandwidth:  2374.954 )
=========================
       0        -1959 
       1        -1679 
       2        -1530 
       3        -1446 
       4        -1412 
       5        -1412 
Fixed bandwidth: 2374.954 AICc value: 4755.482 
 Iteration    Log-Likelihood:(With bandwidth:  2377.371 )
=========================
       0        -1959 
       1        -1680 
       2        -1531 
       3        -1447 
       4        -1413 
       5        -1413 
Fixed bandwidth: 2377.371 AICc value: 4755.48 
 Iteration    Log-Likelihood:(With bandwidth:  2377.942 )
=========================
       0        -1960 
       1        -1680 
       2        -1531 
       3        -1448 
       4        -1414 
       5        -1414 
Fixed bandwidth: 2377.942 AICc value: 4755.48 
 Iteration    Log-Likelihood:(With bandwidth:  2377.018 )
=========================
       0        -1959 
       1        -1680 
       2        -1531 
       3        -1447 
       4        -1413 
       5        -1413 
Fixed bandwidth: 2377.018 AICc value: 4755.48 
bw_new.fixed
[1] 2377.371

We can see now that the revised fixed bandwidth is 2,377.371m.

gwlr_new.fixed <- ggwr.basic(status ~ 
                 distance_to_tertiary_road +
                 distance_to_city +
                 distance_to_town + 
                 water_point_population+ 
                 local_population_1km +
                 usage_capacity +
                 is_urban +
                 water_source_clean,
              data = Osun_wp_sp_new,
              bw=bw_new.fixed,
              family="binomial",
              kernel = "gaussian",
              adaptive=FALSE,
              longlat=FALSE)
 Iteration    Log-Likelihood
=========================
       0        -1959 
       1        -1680 
       2        -1531 
       3        -1447 
       4        -1413 
       5        -1413 
gwlr_new.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-17 21:44:49 
   Call:
   ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city + 
    distance_to_town + water_point_population + local_population_1km + 
    usage_capacity + is_urban + water_source_clean, data = Osun_wp_sp_new, 
    bw = bw_new.fixed, family = "binomial", kernel = "gaussian", 
    adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_tertiary_road distance_to_city distance_to_town water_point_population local_population_1km usage_capacity is_urban water_source_clean
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-129.368    -1.750     1.074     1.742    34.126  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.540e-01  1.055e-01   3.354 0.000796
distance_to_tertiary_road                 1.001e-04  2.040e-05   4.910 9.13e-07
distance_to_city                         -1.764e-05  3.391e-06  -5.202 1.97e-07
distance_to_town                         -1.544e-05  2.825e-06  -5.466 4.60e-08
water_point_population                   -5.098e-04  4.476e-05 -11.390  < 2e-16
local_population_1km                      3.452e-04  1.779e-05  19.407  < 2e-16
usage_capacity1000                       -6.206e-01  6.966e-02  -8.908  < 2e-16
is_urbanTRUE                             -2.667e-01  7.474e-02  -3.569 0.000358
water_source_cleanProtected Shallow Well  4.947e-01  8.496e-02   5.823 5.79e-09
water_source_cleanProtected Spring        1.279e+00  4.384e-01   2.917 0.003530
                                            
Intercept                                ***
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
water_point_population                   ***
local_population_1km                     ***
usage_capacity1000                       ***
is_urbanTRUE                             ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.9  on 4746  degrees of freedom
AIC: 5708.9

Number of Fisher Scoring iterations: 5


 AICc:  5708.923
 Pseudo R-square value:  0.129406
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2377.371 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -3.7021e+02 -4.3797e+00  3.5590e+00
   distance_to_tertiary_road                -3.1622e-02 -4.5462e-04  9.1291e-05
   distance_to_city                         -5.4555e-02 -6.5623e-04 -1.3507e-04
   distance_to_town                         -8.6549e-03 -5.2754e-04 -1.6785e-04
   water_point_population                   -2.9696e-02 -2.2705e-03 -1.2277e-03
   local_population_1km                     -7.7730e-02  4.4281e-04  1.0548e-03
   usage_capacity1000                       -5.5889e+01 -1.0347e+00 -4.1960e-01
   is_urbanTRUE                             -7.3554e+02 -3.4675e+00 -1.6596e+00
   water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01  6.2378e-01
   water_source_cleanProtected.Spring       -1.3630e+03 -5.3436e+00  2.7714e+00
                                                3rd Qu.      Max.
   Intercept                                 1.3755e+01 2171.6373
   distance_to_tertiary_road                 6.3011e-04    0.0237
   distance_to_city                          1.5921e-04    0.0162
   distance_to_town                          2.4490e-04    0.0179
   water_point_population                    4.5879e-04    0.0765
   local_population_1km                      1.8479e-03    0.0333
   usage_capacity1000                        3.9113e-01    9.2449
   is_urbanTRUE                              1.0554e+00  995.1840
   water_source_cleanProtected.Shallow.Well  1.9564e+00   66.8914
   water_source_cleanProtected.Spring        7.0805e+00  208.3749
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2815.659 
   AIC : 4418.776 
   AICc : 4744.213 
   Pseudo R-square value:  0.5691072 

   ***********************************************************************
   Program stops at: 2022-12-17 21:45:44 

We can see that in this scenario, the AIC value has slightly increased to 4418.776 and the AICc value has slightly decreased to 4744.213. The pseudo r-squared value has also decreased slightly to 0.569.

Usually, the performance would have dropped significantly with a reduction in explanatory variables, but it does not happen in this case. Hence, we can prove that the geographically weighted logistic regression model is still a more robust measure, as it did not show much changes in the AIC from the previous geographically weighted logistic regression model, despite having dropped some of the explanatory variables.

Model Assessment

Converting SDF into sf data frame

To assess the performance of the gwLR, firstly, we will convert the SDF object in as data frame by using the code chunk below.

gwr_new.fixed <- as.data.frame(gwlr_new.fixed$SDF)

Next, we will label yhat values greater than or equal to 0.5 into 1 and else, 0. The result of the logic comparison operation will be saved into a field called most. most is a logical function that has only True or False values, which matches with the observed values.

gwr_new.fixed <- gwr_new.fixed %>%
  mutate(most = ifelse(
    gwr_new.fixed$yhat >= 0.5, T, F))

The code chunk below generates another confusion matrix, which will be used in comparison to the confusion matrix generated at an earlier step (global version).

gwr_new.fixed$y <- as.factor(gwr_new.fixed$y)
gwr_new.fixed$most <- as.factor(gwr_new.fixed$most)
CM_new <- confusionMatrix(data=gwr_new.fixed$most,
                          positive="TRUE",
                      reference = gwr_new.fixed$y)

CM_new
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1833  268
     TRUE    281 2374
                                          
               Accuracy : 0.8846          
                 95% CI : (0.8751, 0.8935)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7661          
                                          
 Mcnemar's Test P-Value : 0.6085          
                                          
            Sensitivity : 0.8986          
            Specificity : 0.8671          
         Pos Pred Value : 0.8942          
         Neg Pred Value : 0.8724          
             Prevalence : 0.5555          
         Detection Rate : 0.4992          
   Detection Prevalence : 0.5582          
      Balanced Accuracy : 0.8828          
                                          
       'Positive' Class : TRUE            
                                          

As we can see, all 3 models have more or less the structure for the confusion matrix generated. The accuracy in the revised model has increased from 0.8837 to 0.8846, which is a slight increase.

Most of the other values did not vary much from the previous geographically weighted logistic regression model, which shows that overall, the geographically weighted logistic regression model is a very robust model. It’s results do not vary much despite having some of the explanatory variables being dropped.

Visualising revised gwLR

gwr_sf_new.fixed <- cbind(Osun_wp_sf_selected, gwr_new.fixed)

The code chunk below is used to create an interactive point symbol map.

tmap_mode("view")
tmap mode set to interactive viewing
prob_T_new <- tm_shape(Osun) +
  tm_polygons(alpha=0.1) +
  tm_text(text="ADM2_EN") +
  tm_shape(gwr_sf_new.fixed) +
  tm_dots(col="yhat",
          border.col="gray60",
          border.lwd=1) +
  tm_view(set.zoom.limits = c(8,14))
prob_T_new

As we can see, some of the water points are still relatively closely clustered, and the results are similar to the ones that are plotted in the previous gwLR model.

A side-by-side comparison would show the similarities.

tmap_arrange(prob_T, prob_T_new, asp=1, ncol=2)

Conclusion

As we can see, the geographically weighted logistic regression model is a very robust measure when it comes to modeling the Spatial Variation of the Explanatory Factors of Water Point Status, than as compared to the normal logistic regression model.